Periodicity and stability of a nonlinear periodic integro-differential prey-competition model with infinite delays

A nonlinear integro-differential prey-competition model with infinite delays is proposed. By using the method of coincidence degree and constructing a suitable Lyapunov functional, some sufficient conditions are obtained for the existence of an unique globally attractive strictly positive (componentwise) periodic solution.     

Positive periodic solution of a nonlinear integro-differential prey-competition impulsive model with infinite delays

By using the method of coincidence degree, some conditions are obtained for the existence of positive periodic solution of a nonlinear integro-differential prey-competition impulsive model with infinite delays.     

Non-autonomous nonlinear integro-differential equations with infinite delay

In this paper we consider a non-autonomous abstract nonlinear Volterra integro-differential equation with infinite delay in a Banach space. We establish the existence and uniqueness of a strict solution under a certain local Lipschitz condition on the nonlinear map and an integrability condition on the kernel.     

Permanence and global stability for nonautonomous N-species Lotka-Volterra competitive system with impulses and infinite delays

In this paper, we consider a class of nonautonomous N-species Lotka-Volterra competitive systems with impulses and infinite delays. By developing the methods given in Teng (2002) [25], we give sufficient conditions for permanence and global attractivity of the system.     

On a periodic prey-predator system with infinite delays

§1　IntroductionAnvarovandLarinov[1]introducedthefollowingprey-predatorsystem:x(t)=x(t)[α-γy(t)-γ∫∞0K1(s)y(t-s)ds-∫∞0∫∞0R1(s,θ)y(t-s)y(t-θ)dθds],y(t)=y(t)[-β μx(t) μ∫∞0K2(s)x(t-s)ds ∫∞0∫∞0R2(s,θ)x(t-θ)x(t-s)dθds],(1)whereα,γ,βandμarepositiveconstants,Ki∈C([0,∞),(0,∞))andRi∈C([0,∞)×[0,∞),(0,∞)),i=1,2.Fortheecologicalsenseofsystem(1),wereferto[1,2]andrefer-encescitedtherein.Sincerealisticmodelsrequiretheinclusionoftheeffectofchangingen-vironment,itmot…     

Global asymptotic stability in n-species nonautonomous Lotka–Volterra competitive systems with infinite delays

By means of Lyapunov functional, we have succeeded in establishing the global asymptotic stability of the positive solutions of a delayed n-species nonautonomous Lotka–Volterra type competitive system without dominating instantaneous negative feedbacks. As a corollary, we show that the global asymptotic stability of the positive solution is maintained provided that the delayed negative feedbacks dominate other interspecific interaction effects with delays and the mean delays are sufficiently small.     

Asymptotic stability of impulsive stochastic partial differential equations with infinite delays

In this paper, we study the existence and asymptotic stability in pth moment of mild solutions to nonlinear impulsive stochastic partial differential equations with infinite delay. By employing a fixed point approach, sufficient conditions are derived for achieving the required result. These conditions do not require the monotone decreasing behaviour of the delays.     

Threshold dynamics of a nonlinear multi‐group epidemic model with two infinite distributed delays

The global stability of equilibria is investigated for a nonlinear multi‐group epidemic model with latency and relapses described by two distributed delays. The results show that the global dynamics are completely determined by the basic reproduction number under certain reasonable conditions on the nonlinear incidence rate. Moreover, compared with the results in Michael Y. Li and Zhisheng Shuai, Journal Differential Equations 248 (2010) 1–20, it is found that the two distributed delays have no impact on the global behaviour of the model. Our study improves and extends some known results in recent literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Positive almost periodic solution for a class of Lasota-Wazewska model with infinite delays

By utilizing a fixed theorem in cones, we study the existence of a unique positive almost periodic solution for a generalized Lasota-Wazewska model with infinite delays. Some sufficient conditions which ensure the existence of a unique positive almost periodic solution are derived and it cannot be obtained by the contraction mapping principle. Furthermore, under proper conditions, we establish some criteria to ensure that all solutions of this model converge exponentially to a positive almost periodic solution. An example is provided to illustrate the effectiveness of the proposed result.     

Local existence for nonlinear Volterra integrodifferential equations with infinite delay

In this paper, we investigate the local existence and uniqueness of solutions to integrodifferential equations with infinite delay, which are more general than those in previous studies. We assume that the linear part of the equation is nondensely defined and satisfies a Hille–Yosida condition. Moreover, the continuity of solutions with respect to initial conditions is also studied. In order to illustrate our abstract results, we conclude this work with an example.     

Permanence of a delayed discrete mutualism model with feedback controls

This paper discusses a two species discrete model of mutualism with delays and feedback controls. Sufficient conditions are obtained for the permanence of the system. The results show that feedback control variables have no influence on the persistence property of the system.     

一类具有时滞和反馈控制的非线性离散模型的持久性

研究了一类具有时滞和反馈控制的非线性单种群离散模型的持久性,获得了该系统持久的充分条件,并通过例子表明结果的可行性,所得结果推广了已有文献的相关结果.     

Wavefronts for a global reaction-diffusion population model with infinite distributed delay

We consider a global reaction-diffusion population model with infinite distributed delay which includes models of Nicholson's blowflies and hematopoiesis derived by Gurney, Mackey and Glass, respectively. The existence of monotone wavefronts is derived by using the abstract settings of functional differential equations and Schauder fixed point theory.     

Well-posedness of a nonlinear integro-differential problem and its rearranged formulation

We study the existence and uniqueness of solutions of a nonlinear integro-differential problem which we reformulate introducing the notion of the decreasing rearrangement of the solution. A dimensional reduction of the problem is obtained and a detailed analysis of the properties of the solutions of the model is provided. Finally, a fast numerical method is devised and implemented to show the performance of the model when typical image processing tasks such as filtering and segmentation are performed.     

Controllability of nonlinear integro-differential third order dispersion system

In this short article, sufficient condition for controllability of nonlinear dispersion system is studied. The result is obtained by using the Schaefer fixed-point theorem. This work extends the work of Chalishajar, George and Nandakumaran [D.N. Chalishajar, R.K. George, A.K. Nandakumaran, Exact controllability of the third order nonlinear dispersion equation, J. Math. Anal. Appl. 332 (2007) 1028-1044]. Usually authors assume the compactness of semigroup while studying the controllability. Here we drop this assumption and prove the controllability result.     

Saddle-point equilibrium sequence in one class of singular infinite horizon zero-sum linear-quadratic differential games with state delays

ABSTRACT

We consider an infinite horizon zero-sum linear-quadratic differential game with state delays in the dynamics. The cost functional of this game does not contain a control cost of the minimizing player (the minimizer), meaning that the considered game is singular. For this game, definitions of the saddle-point equilibrium and the game value are proposed. These saddle-point equilibrium and game value are obtained by a regularization of the singular game. Namely, we associate this game with a new differential game for the same equation of dynamics. The cost functional in the new game is the sum of the original cost functional and an infinite horizon integral of the square of the minimizer's control with a small positive weight coefficient. This new game is regular, and it is a cheap control game. An asymptotic analysis of this cheap control game is carried out. Using this asymptotic analysis, the existence of the saddle-point equilibrium and the value of the original game is established, and their expressions are derived. Illustrative example is presented.     

Permanence of a discrete SIRS epidemic model with time delays

In this paper, we consider the permanence of a discrete SIRS epidemic model with time delays. This model is constructed from the discretization by the Euler method. Applying the technique to prove the existence of an eventual lower bound in a continuous epidemic model, a sufficient condition for the permanence of the above discrete SIRS epidemic model is obtained.     

A note on a class of singular integro-differential equations

A simplified analysis is employed to handle a class of singular integro-differential equations for their solutions